(CNN) -- As she takes part in the Republican National Convention with Sen. John McCain, the abuse of power investigation facing Alaska Gov. Sarah Palin at home is charging ahead, and the governor is expected to be questioned this month, according to the lawmaker overseeing the investigation.

Palin's attorney is locked in a legal battle with the Alaska legislature, insisting that it does not have "primary jurisdiction" in the matter. The Democratic chairman of the Alaska Senate Judiciary Committee, meanwhile, has threatened subpoenas if Palin doesn't continue to cooperate in organizing witness interviews.

Letters between the two sides show a legal back-and-forth over several sticking points, with the committee chairman saying that Palin's newfound status as McCain's running mate "does not change the steps" the committee is taking and that he requests a "September date for the governor's deposition."

The inquiry boils down to whether Palin used her power to have a state commissioner fired for refusing to dismiss her former brother-in-law from the state police. She has denied any wrongdoing.

Her sudden, unexpected entry into the presidential race has taken the web of characters involved in the dispute and thrust them into the national spotlight. Watch McCain praise Palin »

The case centers on Walt Monegan, the public safety commissioner who was let go in July. After his dismissal, he said he had felt pressured by the governor's office to fire state trooper Mike Wooten, who had been in a messy divorce and child custody battle with Palin's sister.

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Monegan said members of the governor's staff and her husband, Todd Palin, had called and questioned him about Wooten.

At a news conference last month, Palin acknowledged that her administration had made more than 20 calls to the Department of Public Safety regarding Wooten. However, she said that Wooten had nothing to do with Monegan's firing and that she had simply wanted new leadership to take the Department of Public Safety in "a new direction." The post of commissioner is a gubernatorial appointment.

"The individual inquiries taken by themselves are one thing. Many of these inquiries were completely appropriate; however, the serial nature of the contacts understandably could be perceived as some kind of pressure, presumably at my direction," she said.

Palin revealed a recording of one phone call made by her boards and commissions director, Frank Bailey, to a trooper lieutenant. In the call, Bailey complained that there had been "absolutely no action for a year on this issue" and said that Palin and her husband were frustrated that Wooten was still a state trooper.

To Todd Palin; five children. The oldest, Track, serves in the U.S. Army; the youngest, Trig, has Down syndrome.

Sources: Almanac of American Politics, CNN

"I am truly disappointed and disturbed to learn that a member of this administration contacted the Department of Public Safety regarding Trooper Wooten," Palin said in a statement last month. "At no time did I authorize any member of my staff to do so."

Palin placed Bailey on paid leave until the investigation is over.

The state Legislative Council -- a bipartisan group of lawmakers -- voted in July to spend $100,000 on an investigation and hired Stephen Branchflower, who spent 28 years as a state prosecutor in Anchorage, as special counsel to lead it. The council chose Sen. Hollis French, the chairman of the Judiciary Committee, to manage the investigation.

On Friday, Palin's attorney, Thomas Van Flein, sent a letter via e-mail to Branchflower. "We fully welcome a fair inquiry into these allegations, and believe that the Personnel Board is statutorily mandated to oversee these proceedings," he said, adding that under state law, the Personnel Board has "primary jurisdiction."

The Personnel Board is "an independent agency composed of members appointed by the governor," according to its Web site. Its duties include hearing ethics complaints against Executive Branch employees.

Palin's attorney also wrote that he understood that there was an "informal agreement to share information the parties obtain through their own inquiries." He requested copies of all witness statements, documentary evidence, a witness list and the "complaint or other charging document."

And he requested a meeting with the investigator, Branchflower, to schedule depositions of witnesses.

Van Flein vowed full cooperation and said he would reciprocate in providing information.

Branchflower did not respond to the letter. Instead, French wrote a response Monday, saying the Personnel Board "would not have jurisdiction unless someone filed a complaint."

French asked whether Palin was aware that her attorney seemed to be "challenging" the legislature's jurisdiction. "Such a position is at odds with our state's constitution, and with your client's public statements."

As for the copies of witness statements, French wrote that he had instructed Branchflower not to comply.

"I think you would agree that it would be highly unusual for an investigator to share information with one of the targets of the investigation. I am unaware of any precedent for such an arrangement."

He also said he told the investigator, Branchflower, to stick to his established schedule for interviewing witnesses -- and that any delays "would cause me to convene a meeting of the Judiciary Committee and ask that subpoenas be considered."

Although Palin's "new political role" as Sen. John McCain's running mate "will make it more challenging for her to make time for this investigation," French wrote, her promises to cooperate and her vows that the campaign trail will not interfere with state business "should result in a concrete willingness to schedule and conclude her deposition."

The two letters were first published by the Anchorage Daily News. Van Flein's office then provided copies to CNN.